Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61917 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3207 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,13)(6,10)(7,11)(8,12)(9,15)(14,18)(16,17)(19,20,23,27)(21,28,30,33)(22,26,24,31)(25,32,34,29);(3,5,8,15)(4,9,12,13)(6,14,7,16)(10,17,11,18)(19,20)(21,25)(22,31)(23,27)(24,26)(28,29)(30,34)(32,33);(3,6,8,7)(4,10,12,11)(5,14,15,16)(9,17,13,18)(19,21)(20,25)(22,29)(23,30)(24,32)(26,33)(27,34)(28,31);(3,7,8,6)(4,11,12,10)(5,14,15,16)(9,17,13,18)(19,22,23,24)(20,26,27,31)(21,29,30,32)(25,33,34,28);(4,12)(9,13)(10,11)(17,18)(20,27)(25,34)(26,31)(28,33)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485789);